Provided by: libmath-planepath-perl_125-1_all bug

NAME

       Math::PlanePath::DiagonalsOctant -- points in diagonal stripes for an eighth of the plane

SYNOPSIS

        use Math::PlanePath::DiagonalsOctant;
        my $path = Math::PlanePath::DiagonalsOctant->new;
        my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

       This path follows successive diagonals downwards from the Y axis down to the X=Y centre
       line, traversing the eighth of the plane on and above X=Y.

           8 |  21 27 33 40 47 55 63 72 81
             |    \  \  \  \  \  \  \
           7 |  17 22 28 34 41 48 56 64
             |    \  \  \  \  \  \
           6 |  13 18 23 29 35 42 49
             |    \  \  \  \  \
           5 |  10 14 19 24 30 36
             |    \  \  \  \
           4 |   7 11 15 20 25
             |    \  \  \
           3 |   5  8 12 16
             |    \  \
           2 |   3  6  9
             |    \
           1 |   2  4
             |
         Y=0 |   1
             + ----------------------------
               X=0  1  2  3  4  5  6  7  8

       N=1,4,9,16,etc on the X=Y leading diagonal are the perfect squares.  N=2,6,12,20,etc at
       the ends of the other diagonals are the pronic numbers k*(k+1).

       Incidentally "octant" usually refers to an eighth of a 3-dimensional coordinate space.
       Since "PlanePath" is only 2 dimensions there's no confusion and at the risk of abusing
       nomenclature half a quadrant is reckoned as an "octant".

   Pyramid Rows
       Taking two diagonals running from k^2+1 to (k+1)^2 is the same as a row of the step=2
       "PyramidRows" (see Math::PlanePath::PyramidRows).  Each endpoint is the same, but here
       it's two diagonals instead of one row.  For example in the "PyramidRows" the Y=3 row runs
       from N=10 to N=16 ending at X=3,Y=3.  Here that's in two diagonals N=10 to N=12 and then
       N=13 to N=16, and that N=16 endpoint is the same X=3,Y=3.

   Direction
       Option "direction => 'up'" reverses the order within each diagonal and counts upward from
       the centre to the Y axis.

           8 |  25 29 34 39 45 51 58 65 73
             |    \  \  \  \  \  \  \
           7 |  20 24 28 33 38 44 50 57
             |    \  \  \  \  \  \
           6 |  16 19 23 27 32 37 43
             |    \  \  \  \  \
           5 |  12 15 18 22 26 31
             |    \  \  \  \
           4 |   9 11 14 17 21             direction => "up"
             |    \  \  \
           3 |   6  8 10 13
             |    \  \
           2 |   4  5  7
             |    \
           1 |   2  3
             |
         Y=0 |   1
             +---------------------------
               X=0  1  2  3  4  5  6  7  8

       In this arrangement N=1,2,4,6,9,etc on the Y axis are alternately the squares and the
       pronic numbers.  The squares are on even Y and pronic on odd Y.

   N Start
       The default is to number points starting N=1 as shown above.  An optional "n_start" can
       give a different start, in the same diagonals sequence.  For example to start at 0,

           n_start => 0                    n_start=>0
           direction => "down"             direction=>"up"

             6  | 12                        | 15
             5  |  9 13                     | 11 14
             4  |  6 10 14                  |  8 10 13
             3  |  4  7 11 15               |  5  7  9 12
             2  |  2  5  8                  |  3  4  6
             1  |  1  3                     |  1  2
           Y=0  |  0                        |  0
                +--------------             +--------------
                 X=0  1  2  3                X=0  1  2  3

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

       "$path = Math::PlanePath::DiagonalsOctant->new ()"
       "$path = Math::PlanePath::DiagonalsOctant->new (direction => $str, n_start => $n)"
           Create and return a new path object.

       "($x,$y) = $path->n_to_xy ($n)"
           Return the X,Y coordinates of point number $n on the path.

           For "$n < 0.5" the return is an empty list, it being considered the path begins at 1.

       "$n = $path->xy_to_n ($x,$y)"
           Return the point number for coordinates "$x,$y".  $x and $y are each rounded to the
           nearest integer, which has the effect of treating each point $n as a square of side 1.

       "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
           The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in
           the rectangle.

FORMULAS

   N to X,Y
       To break N into X,Y it's convenient to take two diagonals at a time, since the length then
       changes by 1 each pair making a quadratic.  Starting at each X=0,Y=odd just after perfect
       square N allows just a sqrt.

           Nstart = d*d+1

       where d numbers diagonal pairs, eg. d=3 for X=0,Y=5 going down.  This is easily reversed
       as

           d = floor sqrt(N-1)

       The code reckons the start of the diagonal as 0.5 further back, so that N=9.5 is at
       X=-.5,Y=5.5.  To do that d is formed as

           d = floor sqrt(N-0.5)
             = int( sqrt(int(4*$n)-2)/2 )

       Taking /2 out of the sqrt helps with "Math::BigInt" which circa Perl 5.14 doesn't inter-
       operate with flonums very well.

       In any case N-Nstart is an offset into two diagonals, the first of length d many points
       and the second d+1.  For example d=3 starting Y=5 for points N=10,11,12 followed by Y=6
       N=13,14,15,16.

       The formulas are simplified by calculating a remainder relative to the second diagonal, so
       it's negative for the first and positive for the second,

           Nrem = N - (d*(d+1)+1)

       d*(d+1)+1 is 1 past the pronic numbers when end each first diagonal, as described above.
       In any case for example d=3 is relative to N=13 making Nrem=-3,-2,-1 or Nrem=0,1,2,3.

       To include the preceding 0.5 in the second diagonal simply means reckoning Nrem>=-0.5 as
       belonging to the second.  In that base

           if Nrem >= -0.5
             X = Nrem            # direction="down"
             Y = 2*d - Nrem
           else
             X = Nrem + d
             Y = d - Nrem - 1

       For example N=15 Nrem=1 is the first case, X=1, Y=2*3-1=5.  Or N=11 Nrem=-2 the second
       X=-2+3=1, Y=3-(-2)-1=4.

       For "up" direction the Nrem and d are the same, but the coordinate directions reverse.

           if Nrem >= -0.5
             X = d - Nrem        # direction="up"
             Y = d + Nrem
           else
             X = -Nrem - 1
             Y = 2d + Nrem

       Another way is to reckon Nstart from the X=0,Y=even diagonals, which is then two diagonals
       of the same length and d formed by a sqrt inverting a pronic Nstart=d*(d+1).

   Rectangle to N Range
       Within each row increasing X is increasing N, and in each column increasing Y is
       increasing N.  This is so in both "down" and "up" arrangements.  On that basis in a
       rectangle the lower left corner is the minimum N and the upper right is the maximum N.

       If the rectangle is partly outside the covered octant then the corners must be shifted to
       put them in range, ie. trim off any rows or columns entirely outside the rectangle.  For
       the lower left this means,

             |  |    /
             |  |   /
             +--------          if x1 < 0 then x1 = 0
           x1   | /             increase x1 to within octant
                |/
                +

                |  |/
                |  |            if y1 < x1 then y1 = x1
                | /|            increase y1 to bottom-left within octant
                |/ +----y1
                +  x1

       And for the top right,

                |    /  x2
                | ------+ y2    if x2 > y2 then x2 = y2
                |  /    |       decrease x2 so top-right within octant
                | /     |         (the end of the y2 row)
                |/
                +

OEIS

       Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

           <http://oeis.org/A055087> (etc)

           direction=down
             A002620    N at end each run X=k,Y=k and X=k,Y=k+1
           direction=down, n_start=0
             A055087    X coord, runs 0 to k twice
             A082375    Y-X, runs k to 0 or 1 stepping by 2
             A005563    N on X=Y diagonal, X*(X+2)

           direction=up
             A002620    N on Y axis, end of each run, quarter squares
           direction=up, n_start=0
             A024206    N on Y axis (starting from n=1 is Y=0, so Y=n-1)
             A014616    N in column X=1 (is Y axis N-1, from N=3)
             A002378    N on X=Y diagonal, pronic X*(X+1)

           either direction, n_start=0
             A055086    X+Y, k repeating floor(k/2)+1 times

           A004652      N start and end of each even-numbered diagonal

           permutations
             A056536     N of PyramidRows in DiagonalsOctant order
             A091995      with DiagonalsOctant direction=up
             A091018      N-1, ie. starting from 0
             A090894      N-1 and DiagonalsOctant direction=up

             A056537     N of DiagonalsOctant at X,Y in PyramidRows order
                          inverse of A056536

SEE ALSO

       Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::DiagonalsAlternating,
       Math::PlanePath::PyramidRows

HOME PAGE

       <http://user42.tuxfamily.org/math-planepath/index.html>

LICENSE

       Copyright 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde

       This file is part of Math-PlanePath.

       Math-PlanePath is free software; you can redistribute it and/or modify it under the terms
       of the GNU General Public License as published by the Free Software Foundation; either
       version 3, or (at your option) any later version.

       Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY
       WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
       PURPOSE.  See the GNU General Public License for more details.

       You should have received a copy of the GNU General Public License along with Math-
       PlanePath.  If not, see <http://www.gnu.org/licenses/>.